Don't worry, Carol, it is not a fungus, although those leaf spots do mimic fungal diseases. Cordylines are very sensitive to minerals salts that can build up in soil and cause leaf spotting. Fluoride is one mineral salt, but not normally a problem at the concentration levels added to water supplies. However, hard water is high in various mineral salts and can cause Cordyline leaf spotting. Using too much fertilizer or perlite can also affect this plant. Finally, be aware that if the soil becomes too dry, the concentration of any mineral salts in the soil will increase. That doesn't argue for keeping the soil constantly wet, but it does mean it is important to never let the soil get too dry.
Frankly, it is hard to avoid some leaf spotting with potted Cordylines, especially on older leaves. Use filtered or distilled water if you have a water quality problem. Skip fertilizer altogether. And water your plant thoroughly as soon as the surface of the soil feels almost dry.
